Quantization scheme for modular q-difference equations

Modular pairs of some second order q-difference equations are considered. These equations may be interpreted as a quantum mechanics of a sort of hyperelliptic pendulum. It is shown the quantization of a spectrum may be provided by the condition of the analyticity of the wave function. Baxter’s t-Q equations for the quantum relativistic Toda chain in the ``strong coupling regime’’ are related to the system considered, and the quantization condition for Q-operator is also considered.